Student Research Project-Raul Aviles Poblador

FAKULTÄT ELEKTROTECHNIKUND INFORMATIONSTECHNIK

Institut für Nachrichtentechnik

Vodafone Chair

Mobile Communication Systems

Analysis of Performance-Complexity Trade-off in

Iterative Detection-Decoding

STUDIENARBEIT

STUDENT RESEARCH PROJECT

Autor: Raúl Avilés Poblador

Supervisors: Dipl.-Ing. Esther Pérez Adeva, Dipl.-Ing. Steffen Kunze

Date: March 30th, 2012

Selbstständigkeitserklärung

Hiermit erkläre ich, dass ich die am heutigen Tag beim Prüfungsausschuss der FakultätElektrotechnik und Informationstechnik eingereichte Studienarbeit zum Thema

Analysis of Performance-Complexity Trade-off in Iterative Detection-Decoding

vollkommen selbstständig von mir verfasst, und keine anderen als die angegebenenQuellen und Hilfsmittel verwendet sowie Zitate kenntlich gemacht wurden.

Dresden, den 30. März 2012

Raúl Avilés Poblador

iii

Aknowledgements

I would like to express my deepest gratitude to Esther, my tutor on this project,who has always trusted me and has given an incredibly valuable input on my work.Without her ideas and advice this project would not have been the same. Moreover,I would like to thank Dipl.-Ing. Steffen Kunze, Prof. Dr.-Ing. Gerhard Fettweis andthe whole Vodafone Chair Mobile Communications Systems at Technische UniversitätDresden for allowing me to develop this project.

No habría sido capaz de llegar hasta aquí sin el inestimable apoyo y amor de mis padres,o la sonrisa de mis hermanos. Ellos son los principales culpables de que consiga loque pueda conseguir, porque siempre han creído en mí, por muy locas que fueran misideas o mis planes.

Además, hay una persona en mi vida que ha sabido encajar de una forma especialconmigo. Gracias a ella las largas horas de trabajo se hacen menos pesadas sabiendoque al final del día va a estar ahí. No hay suficientes palabras para describir lo felizque me haces.

Por último, pero no menos importante, quiero agradecer su constancia, amistad ycompañía a todos los amigos, que desde lejos o cerca forman una parte indispensablede mi vida.

A todos vosotros, gracias.

To all of you, thank you.

v

Abstract

In today’s and in future further evolved 4G wireless standards, such as WiMAX,3GPP-LTE or LTE-Advanced, receiver terminals have to support numerous operatingmodes for each protocol as well as sophisticated transmission techniques, especiallyenhanced multiple-input multiple-output (MIMO) detection and iterative forward er-ror correction (FEC). These two belong to the most computationally intensive partsof the receiver-side baseband signal processing chain. Motivated by the tremendousgain in FEC, the turbo principle has been also extended to include iterations betweenMIMO detector and channel decoder. By exchanging soft information in iterativefashion, the quality of wireless transmission can be significantly improved. However,this further increases the receiver computational complexity, especially concerning thedetector module. In this regard, it becomes critical that both detector and FEC areable to interact in a flexible and efficient way, not compromising the challenging high-throughput and flexibility requirements associated with 4G standards. Main goal ofthis student project is thus the investigation of the performance-complexity trade-off in iterative detection↔decoding systems, for different system configurations. Inparticular, the project covers:

• Investigation of MIMO detection and FEC strategies.

• Exploration, via Matlab simulations, of performance-complexity trade-off fordifferent transmission and receiver system configurations.

• Interpretation and reporting of obtained results, including identification of re-ceiver configurations providing the best trade-off for each considered scenario.

vii

Contents

Aknowledgements iv

Abstract vi

List of Figures x

List of Tables xi

Abbreviations, Symbols and Mathematical Notation xii

1 Introduction 1

2 Fundamentals 32.1 Communications System Model . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Iterative Detection-Decoding . . . . . . . . . . . . . . . . . . . 42.1.1.1 Internal and External Iterations . . . . . . . . . . . . 52.1.1.2 Log-Likelihood-Ratio Values . . . . . . . . . . . . . . 5

2.2 MIMO Sphere Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2.1 Tree Search Detection Basics . . . . . . . . . . . . . . . . . . . 62.2.2 Sphere Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.3 Radius and Candidates Determination . . . . . . . . . . . . . . 8

2.3 Forward Error Correction (FEC) . . . . . . . . . . . . . . . . . . . . . 92.4 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Performance-Complexity Trade-off in Iterative Detection-Decoding 133.1 Case Study Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.1 Iterations Configuration . . . . . . . . . . . . . . . . . . . . . . 153.2.1.1 Internal-External Iterations Relationship . . . . . . . 20

3.2.2 MIMO Configuration . . . . . . . . . . . . . . . . . . . . . . . . 203.2.3 Constellation Size . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.4 Decoder Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.5 Interleaver Type . . . . . . . . . . . . . . . . . . . . . . . . . . 27

viii

3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Adaptive LLR Clipping 314.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.2 Algorithm and Implementation . . . . . . . . . . . . . . . . . . . . . . 33

4.2.1 LLR clipping value initialization . . . . . . . . . . . . . . . . . 344.2.2 Estimation of the previous frame BER . . . . . . . . . . . . . . 354.2.3 Comparison of the current frame BER with the TER to calculate

a new LLR clipping value . . . . . . . . . . . . . . . . . . . . . 364.2.4 Passing the clipping value to the Sphere Detector and clip the

La(c’) values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Summary 435.1 Outcome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.2 Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

A Performance-Complexity Data Charts 47A.1 MIMO 2x2, 64 QAM, PCCC Decoder . . . . . . . . . . . . . . . . . . 48A.2 MIMO 2x2, 64 QAM, LDPC Decoder . . . . . . . . . . . . . . . . . . 49A.3 MIMO 4x4, 64 QAM, PCCC Decoder . . . . . . . . . . . . . . . . . . 50A.4 MIMO 4x4, 64 QAM, LDPC Decoder . . . . . . . . . . . . . . . . . . 51A.5 MIMO 4x4, 16 QAM, PCCC Decoder . . . . . . . . . . . . . . . . . . 52A.6 MIMO 4x4, 16 QAM, LDPC Decoder . . . . . . . . . . . . . . . . . . 53

B Performance-Complexity Graphics 54B.1 MIMO 2x2, 64 QAM, PCCC Decoder . . . . . . . . . . . . . . . . . . 54B.2 MIMO 2x2, 64 QAM, LDPC Decoder . . . . . . . . . . . . . . . . . . 57B.3 MIMO 4x4, 64 QAM, PCCC Decoder . . . . . . . . . . . . . . . . . . 60B.4 MIMO 4x4, 64 QAM, LDPC Decoder . . . . . . . . . . . . . . . . . . 63B.5 MIMO 4x4, 16 QAM, PCCC Decoder . . . . . . . . . . . . . . . . . . 65B.6 MIMO 4x4, 16 QAM, LDPC Decoder . . . . . . . . . . . . . . . . . . 68

Bibliography 71

ix

List of Figures

2.1 Comunications System Model . . . . . . . . . . . . . . . . . . . . . . . 32.2 BPSK 4x4 MIMO tree search example . . . . . . . . . . . . . . . . . . 8

3.1 Comparison of complexity of two system configurations . . . . . . . . . 163.2 Eb/N0 at a BER of 10−5, with 8 internal iterations, for a different

number of external iterations, for different system configurations . . . 193.3 Relationship between internal and external iterations for the optimal

configurations (lowest SNR at BER=10−5) in different system configu-rations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Comparison of performance and complexity for different interleavertypes and block sizes for a MIMO 4x4 64 QAM PCCC system . . . . . 28

4.1 Analysis of BER vs. Complexity of two transmission systems with QAMmodulations, MIMO 4x4, Sphere Detector, PCCC Decoder, 1 internaliteration, 1-5 external iterations, clipping value=4.4 . . . . . . . . . . . 32

4.2 Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Comunications System Model with Adaptive LLR Clipping . . . . . . 334.4 Adaptive LLR Clipping Simulation for 4x4 64 QAM PCCC 1 int.it. 1

ext.it., SNR=20 dB, TER=10−2 . . . . . . . . . . . . . . . . . . . . . 39

x

List of Tables

3.1 Detector complexity increase in iterative cases with regard to non-iterative cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Summary of all the data charts in Appendix A. In grey the configura-tions with best performance-complexity trade-offs . . . . . . . . . . . . 17

3.3 Maximum achievable SNR improvement, accomplished with 4 externaliterations, for different system configurations . . . . . . . . . . . . . . 18

3.4 Relationship between the number of external and internal iterations inan optimal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Relative SNR improvement between MIMO 2x2 64 QAM PCCC and4x4 64 QAM PCCC performing 1 external iteration for a different num-ber of internal iterations . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.6 Relative SNR improvement between MIMO 2x2 64 QAM LDPC and4x4 64 QAM LDPC performing 1 external iteration for a different num-ber of internal iterations . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.7 SNR improvement with 4 external iterations compared with only 1 ex-ternal iteration, for 64 QAM and 16 QAM systems . . . . . . . . . . . 23

3.8 Summary of Table 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.9 Relative SNR gain between the SNR iterative improvement (gain of

iterative configurations compared with the non-iterative ones) between4x4 16 QAM and 4x4 64 QAM systems, using PCCC and LDPC . . . 25

3.10 SNR gain and complexity increase in iterative cases with regard to non-iterative cases, for different decoders . . . . . . . . . . . . . . . . . . . 26

3.11 Number of extended nodes for different system configurations, at 10internal iterations and 4 external iterations . . . . . . . . . . . . . . . 27

xi

Abbreviations, Symbols andMathematical Notation

Symbols

c Vector of transmitted bits

∆r Relative distance between the received signal y′′′ and the receivedestimated symbol x̂r

Eb Average bit energy

H Channel matrix

L-value Log-Likelihood Ratio value

La(c) A-Priori Information (Log-Likelihood Ratio) of the detector

L(c) A-Posteriori Information (Log-Likelihood Ratio) of the detector

Le(c) Extrinsic Information (Log-Likelihood Ratio) of the detector

La(c′) A-Priori Information (Log-Likelihood Ratio) of the decoder

Le(u) A-Posteriori Information (Log-Likelihood Ratio) of the decoder

Le(c′) Extrinsic Information (Log-Likelihood Ratio) of the decoder

Lclip LLR clipping value

L0cl LLR initialization clipping value

LTER LLR initialization clipping value

|Lmin| Minimum possible LLR magnitude determined by the fixed pointaccuracy

λ Metric value

λi Partial metric corresponding to layer i

λmin Minimum metric in the candidate list

λmax Maximum metric in the candidate list

m Frame

n Complex noise vector

xii

NR Number of receive antennas

NT Number of transmit antennas

N0 Noise power

Q Unitary matrix of the QR-decomposition

Q Number of constellation symbols

Qs√Q-1

R Upper triangular matrix of the QR-decomposition

R Radius of the search sphere

Rc Coding rate

Rclipped Clipped radius

Runclipped Unclipped radius

s̃ Estimated sent symbol vector

u Vector of information bits

x Sent symbol vector

y Received symbol vector

y′ Received symbol vector, multiplied by QH

y′′ Interference-reduced received symbol vector

y′′′ Interference-reduced received symbol vector, normalized with rii

Mathematical Notation

< Real component

= Imaginary component

‖ · ‖2 Squared Euclidean norm

Abbreviations

BCJR Bahl, Cocke, Jelinek, Raviv-Algorithm

BICM Bit-Interleaved Coded Modulation

BER Bit Error Rate

BPSK Binary Phase Shift Keying

FEC Forward Error Correction

ext.it. External Iterations

xiii

int.it Internal Iterations

LDPC Low-Density Parity-Check

LS Leaf Sequence

LSD List Sphere Detector

MIMO Multiple Input/ Multiple Output

ML Maximum Likelihood

PCCC Parallel Concatenated Convolutional Code

QAM Quadrature Amplitude Modulation

SD Sphere Detector

SISO Soft Input/ Soft Output

SNR Signal to Noise Ratio

STS Single Tree Search

TER Target Error Rate

TS Tuple Search

TSD Tuple Search Detection

xiv

1 Introduction

It is well known that the information era could not be the same without mobile com-munications systems. The second generation of mobile terminals has already beenleft behind, and the third generation is well established in the telecommunicationsmarket. The desire, and more than ever, the need to achieve higher data rateshas become the number one target in the related wireless communications researchfields.

It is in this situation where the 4G wireless standards are everyday more establishedinto the common repertory of technologies accounted by most telecommunication in-dustry leaders. WiMAX, 3GPP-LTE or LTE-Advanced provide an outstanding per-formance when it comes to spectral efficiency and data rates. However, in order tomake this features real, big challenges have to be faced.

Multiple-input multiple-output (MIMO) detection, as well as iterative forward errorcorrection codes— transmission techniques supported by many of the aforementionedstandards, are among the most computationally intensive parts of the receiver-sidebaseband signal processing chain.

The iterative detection↔ decoding approach, which interchanges soft information be-tween the turbo decoder and the sphere detector, improves the communication perfor-mance at the receiver end. Nevertheless the computational complexity is a major sideeffect. Therefore, analysing trade offs between the performance and the complexityof the different transmission techniques combined with several MIMO configurationsand FEC strategies will be the focus of this study.

The fundamentals explained in Chapter 2 will be widely used to interpret the simula-tion results obtained in Chapter 3, where several transmission configurations will beanalysed in order to find trade-offs in performance-complexity regarding the signal-to-noise-ratio needed.

The Chapter 4 will develop the implementation of an adaptive approach of adjustingthe log-likelihood-ratios, which allows to achieve a certain TER. This becomes a rele-vant tool in the iterative detection↔ decoding scheme considered in this study.

1

The results obtained through this study will be sumarised in Chapter 5, and fu-ture work to continue the investigation into the abovementioned topics will be sug-gested.

2

2 Fundamentals

The theoretical concepts needed to analyse and interpret the results of the simulationsand implementations carried out in this work, and to explain the motivation of thisstudy, will be next introduced.

2.1 Communications System Model

For comparability purposes with other related publications, a communications systemmodel similar to the ones used in [Zim07] [Ade09] [Sha11], has been considered, where aMIMO system model is constituted of NT transmit antennas and NR receive antennas,based on a BICM1 system with a QAM2 modulation.

A vector u of independent and identically distributed (i.i.d.) information bits is en-coded by an outer channel code with rate Rc. Two different forward error correction

1Bit Interleaved Coded Modulation: the information sequence is encoded, interleaved, modulatedand then directly mapped onto the transmit antennas. No channel knowledge is required at thetransmitter side.

2Quadrature Amplitude Modulation

Binary Data Source Outer Encoder Interleaver

∏Constellation

Mapper

Binary Data Sink Outer Decoder

Deinterleaver∏-1

Detector /Demapper

Interleaver∏

+

+

Transmitter

Receiver

QAM

û

u c' c

Le(u) La(c')

Le(c') La(c)

Le(c)

-

NR

NTi.i.d. Rayleigh flat channel

L(c)

-

Figure 2.1: Comunications System Model

3

2.1. COMMUNICATIONS SYSTEM MODEL

codes have been used throughout this work, PCCC3 and LDPC4, which will be intro-duced in Section 2.3. The coded vector c′ is bit-interleaved and portioned into blocksc of NT · L bits, where L denotes the number of transmit bits per transmit symbol.A block interleaver and a random interleaver will be introduced in Section 2.3. Af-terwards, the corresponding bits c are mapped (e.g. gray mapping) onto complexconstellation symbols x(c) = [x0, ...xNT−1]

T = map(c).

The channel is modeled as a Rayleigh flat fading, uncorrelated channel, where AWGNis added. The received signal y is given by the equation (2.1).

y = Hx + n (2.1)

On the receiver’s side, the detection of the transmitted bits is carried out by a complexvalued Soft-Input Soft-Output (SISO) MIMO detector with detector ↔ decoder iter-ations according to the turbo principle [Hag02]. The characteristics of the considereddetection↔ decoding approach will be discussed in Section 2.1.1.

In this study, block interleavers have been considered. However, some simulationshave also been performed using a random interleaver to compare the BER perfor-mance. While in a block interleaver there is a two dimensional array where the bitsare flipped row and column wise, a random interleaver performs a random (but known)permutation of the bits. In both cases the aim is to create a more uniform distributionof the errors, trying to mitigate, for example, the burst errors. The influence of the in-terleavers in our simulation results will be referred in Chapter 3.

2.1.1 Iterative Detection-Decoding

For the considered system model with coded transmission, it is suboptimal for theMIMO detector and channel decoder to operate separately and only on individualvectors of the received signal [Sha11], therefore an iterative detection ↔ decodingapproach is considered.

In this way, a better system performance is achieved because the detector makesdecisions using the a priori information provided by the channel decoder, and thechannel decoder makes decisions using the log-likelihood information provided by theMIMO detector.

Our receiver consists of a serial concatenation of an inner MIMO detector and an outerchannel decoder, as shown in Figure 2.1, where both modules accept and generate soft

3Parallel Concatenated Convolutional Code4Low-Density Parity Check

4

2.1. COMMUNICATIONS SYSTEM MODEL

information on the bits of the transmitted codeword c. At the detector, the knowl-edge of the received signal, the channel state information, and the a priori informationLa(c) provided by the decoder are used to generate the a posteriori information L(c).The substraction of the a priori information La(c) from the a posteriori informationL(c) is called extrinsic information Le(c). The extrinsic information is deinterleavedand passed to the channel decoder as its a priori information La(c’). Extrinsic in-formation Le(c’) is also generated in the decoder, which is interleaved and passedas a priori information La(c) to the detector, completing a detection ↔ decodingiteration.

2.1.1.1 Internal and External Iterations

Throughout this study, internal and external iterations of our communications systemmodel will be referred. The following meanings are implied:

• External Iteration: it refers to a complete detection ↔ decoding iteration asdiscussed in Section 2.1.1. They will be also referred as SD-FEC iterations.

• Internal Iteration: it refers to the iterations performed inside the decoder[RVH95]. As introduced before, in this study two decoders are employed (LDPCand PCCC), which will be discussed in Section 2.3. The internal iterations willbe also referred as FEC iterations.

2.1.1.2 Log-Likelihood-Ratio Values

In following discussions, the LLR (Log-Likelihood Ratio) or L-values are constantlyreferred. They are a mean to describe the soft-output in the iterative detection↔ de-coding. The L-value of a bit cm,l is defined as follows5

L (cm,l|y)=ln

(P (cm,l = +1|y)

P (cm,l = −1|y)

)≈− 1

N0min

c|cm,l=+1{λ0}+

1

N0min

c|cm,l=−1{λ0} (2.2)

where cm,l = ±1 represents the l-th bit of the symbol sent by them-th antenna, and λ0represents the distance metric for a vector of received symbols y.

5Max-log approximation is considered [Zim07] [Sha11].

5

2.2. MIMO SPHERE DETECTION

The relationship between the three L-values generated in the iterative detection↔ de-coding is

L (cm,l|y) = La (cm,l) + Le (cm,l|y) (2.3)

Further input on how the L-values are involved in the detection is given in Sec-tion 2.2.

The soft information provided by the L-values provide a measure of how reliable is theoutput of the detection and the output of the decoding. This reliability has a directrelationship with the BER achieved by the transmission system.

The L-values contain two types of information:

• The sign: indicates if the received bit is 1 or 0.

• Magnitude: is the reliability of the decision made upon the sign. A highermagnitude indicates more reliability.

In [Int00], a method to estimate the BER in function of the decoder a posterioriinformation Le(u) is introduced. The error probability of the decision on the bit cwith a posteriori LLR (Le(u)) is

Pe =(

1 + e|Le(u)|)−1

(2.4)

In Chapter 4 the motivation of adjustment of the L-values to modify the BER willbe discussed. This adjustment will be referred as external clipping throughout thisstudy.

2.2 MIMO Sphere Detection

The task of generating soft output in the MIMO detector can be transformed into a treesearch problem [Zim07]. In this study, a Sphere Detector (SD) has been considered,which is a depth-first tree-search-based MIMO detection strategy. The fundamentalsof the abovementioned concepts will be discussed in this chapter.

2.2.1 Tree Search Detection Basics

The basic aim of the detector is finding the detection hypotheses. The L-values inthe equation (2.2) are required to determine the most likely sent symbols x(c) withc = argc|cm,l=±1min {λ0} (the detection hypothesis). Nevertheless the detector also

6


has to determine the counter-hypotheses c = argc|cm,l=±1,cm,l 6=cMLm,l

min {λ0} with theirmetrics for each bit in order to calculate the L-values..

To simplify the detection of these symbols, the detection problem is transformed intoa tree search problem using the QR decomposition H = QR of the channel matrix H

[HtB03], where Q ∈ CNR x NT is a unitary and R ∈ CNR x NT an upper triangularmatrix, with real value entries on its main diagonal. Being y′ = QHy the mod-ified received symbols, the euclidean distance in the detection can be expressed asfollows

‖y −Hx‖ = ‖y′ −Rx‖2 (2.5)

The process of successively solving (2.5) for the potentially sent symbols xi withi = (NT − 1) . . . 0 is equivalent to search a tree layer by layer for xi upon the layermetric of the tree, which is expressed in the equation (2.6). Note that the analysisof the metric at layer i considers the interference of the already detected symbols(j = (i+ 1) . . . (NT − 1)).

λi = λi+1︸︷︷︸+

∣∣∣∣ y′′i︸︷︷︸−riix̂i∣∣∣∣2 − N0

2

L∑j=1

ci,jLa(ci,j)︸︷︷︸ (2.6)

metric fromalready

estimatedsymbols

interference reducedsymbol a priori information

y′′i = y′i −NT−1∑j=i+1

rij x̂j (2.7)

The aim of the detection algorithm is the selection and analysis of nodes relevant for(2.2). A depth-first tree search strategy, known as sphere detection, is used in thisstudy.

2.2.2 Sphere Detection

The target of the sphere detection [Poh81] is reducing the search space, in a sphericalfashion around the received symbol, as quickly as possible to keep the number ofcalculations low. The radius R0 of the sphere is a constraint parameter, which restrictsthe search space and is adapted with the distance metric of the found leaf nodes. Thiswill be described in Section 2.2.3.

7


1

2

3

4

5

6

7

8

9

i = 4

i = 3

i = 2

i = 1

i = 0

Root

Parent node of level i = 2

Child nodes of level i = 3

Leaf nodes

Figure 2.2: BPSK 4x4 MIMO tree search example

An example of BPSK 4x4 MIMO search tree is depicted in figure 2.2 together withrelevant notations. The sphere detection algorithm starts at layer i = NT with anunlimited sphere. At each level i the algorithm analyses the child nodes on its partialmetrics λi basis, and selects one that has not been selected so far. Then, the selectednode is extended by analysing its child nodes at layer i − 1. Whenever a leaf isreached (i = 0), the search radius is adapted. The leaf nodes found during the searchare stored in a list to enable the calculation of the soft-output. Whenever all childnodes from a parent node within the sphere are extended, the search goes one layerup and continues the search. When the root node is reached again the search iscompleted.

Assuming an implementation with one extended node per clock cycle, the number ofextended nodes is considered a complexity factor [Ade09] [AB+05] [MFF09] and it willbe analysed throughout this study.

2.2.3 Radius and Candidates Determination

In this study, a TSD (Tuple Search Detection) algorithm is employed [MvBF09], whichis an improved version of the LSD (List Sphere Decoder) [HtB03].

The election of a good R0 is a key factor to reduce the detection complexity [Zim07].If the sphere radius is too large, many nodes will be visited. On the other hand, if R0

is too small, the sphere will be empty and the search has to be restarted with a largerradius. How the radius is calculated will be next discussed.

For the radius determination a search tuple will be employed. A set of the mostpromising candidates (with regard to the metrics) is identified and the search sphere

8

2.3. FORWARD ERROR CORRECTION (FEC)

is determined over a T -tuple T := {λ0(c1), λ0(c2), . . . , λ0(cT )} of the identified can-didates with lowest λ0. The sphere radius is defined to the maximum tuple met-ric:

R = maxct|ct∈T

{λmin,t} (2.8)

The list is initialised with λ0,t = ∞. The sphere search finds all leaves within thecurrent sphere. Each leaf represents a new element of the tuple, which replaces theelement with worst metrics in the tuple. The replacement occurs only in the case abetter metric is found. This way, only a subset of nodes will remain within the searchsphere.

When it comes to L-values determination, the number of node extensions is very high[MvBF09], because computing the L-values in (2.2) requires the determination of adetection hypothesis and all counter-hypotheses as introduced in Section 2.2.1. There-fore, the combination of the usage of the information gathered during the search incombination with an adapted search strategy becomes of relevance, and the retainmentof leaf candidates in a candidate list is considered.

As a consequence of the radius reduction, less nodes are explored, therefore lesscounter-hypotheses are found and the detection accuracy given by the L-values willbe smaller. Trade-offs between the radius value and the detection accuracy will bediscussed in Chapter 4. The adjustment of the search radius by clipping it, will bereferred as internal clipping throughout this study.

2.3 Forward Error Correction (FEC)

Two common FEC strategies in 4G wireless systems are considered in our communi-cations system model: Parallel Concatenated Convolutional Code (PCCC) and Low-Density Parity Check (LDPC). LDPC is a linear block code whose parity-check matrixis sparse, i.e. it contains only a small number of 1s per row or column. The PCCCturbo codes are built from 2 convolutional codes (CCs). The turbo principle reflectsthat the soft output of one component code (one sub-iteration or half-iteration) is usedas input for the other component code. Two sub-iterations constitute one decoder it-eration (also referred as internal iteration through this work). The iterative decodingis based on symbol-by-symbol SISO decoding of the component codes, where a BCJRalgorithm evaluates the a posteriori probabilities [AN07].

The aforementioned codes are widely used in advanced mobile communications systemsbecause they are capable of achieving bit error rates (BER) as small as 10−5 at a code

9

2.4. SIMULATION SETUP

rate of R = 1/2 and a signal-to-noise ratio Eb/N0 of just 0.7 dB above the Shannonlimit [BGT93]. However, in order to achieve this level of performance, large blocksizes are required. In Chapter 4 the limitations introduced by the block size will bediscussed.

2.4 Simulation Setup

During our simulations, several parameters can be configured to vary the characteris-tics of our communications system model. Relevant simulation parameters consideredin this work, which will be further discussed, are:

• Constellation size: 4 QAM, 16 QAM, 64 QAM

• MIMO configuration (transmit antennas x receive antennas):

– 2 x 2

– 4 x 4

• Iterations configuration:

– Number of external (detection ↔ decoding) iterations

– Number of internal (decoder) iterations

• Clipping Value: a reference value of 4.4 [MvBF09] will be used for all simulationsin Chapter 3

• Type of Decoder (both with a code rate of Rc = 1/2):

– PCCC

– LDPC

• Type of Interleaver:

– Block

– Random

• Interleaver length:

– 232800 bits using PCCC

– 153600 bits using LDPC

10

2.4. SIMULATION SETUP

Through these simulations, trade-offs in performance and complexity will be inves-tigated in Chapter 3. As introduced before, the performance will be referred to thesystem BER and the complexity to the number of nodes extended during the treesearch. Both parameters will be analysed in combination with the signal-to-noiseratio of the system.

In Chapter 4, an implementation on how to adaptively adjust the BER by exploitingthe decoder a posteriori information will be discussed.

11

3 Performance-Complexity Trade-off inIterative Detection-Decoding

The iterative detection ↔ decoding in MIMO systems has been considered in relatedtechnical publications as a mean to improve the system error rate performance byapplying the turbo principle to the exchange of soft information between a MIMOdetector and a channel decoder, for example in [MvBF09], [WM10] and [Sha11].However, the employment of the iterative detection ↔ decoding approach also in-curs computational complexity increases. Some studies have been made on how thenumber of detection ↔ decoding iterations affects the performance and complexityof a system [SBB08] [YJ09] [OM04] [ZZZX07]. In this work, an exhaustive analysisof performance-complexity trade-offs in different system configurations will be tack-led.

Moreover, in the aforementioned publications the dependent relationship between thenumber of decoder iterations and the number of detection ↔ decoding iterations hasnot been considered. An approximated dependency between these two system config-uration parameters will be analysed in Section 3.2.1.1.

3.1 Case Study Setup

Throughout this study, six communication system configurations have been anal-ysed:

• MIMO 2x2, 64 QAM, PCCC Decoder

• MIMO 2x2, 64 QAM, LDPC Decoder





13

3.1. CASE STUDY SETUP

Remain of the simulation setup parameters have been described already in Section 2.4.

For each of these configurations, the system has been simulated with 1 to 10 internal it-erations, and for each number of internal iterations, the system has also been simulatedwith 1 to 5 external iterations (detection↔ decoding iterations).

Totaly, more than 300 simulations have been performed. In order to analyse theperformance-complexity trade-offs, the simulation results have been compiled in twoways, which will constitute the source for this chapter’s conclusions.

Before the beginning of the analysis, the results format will be introduced so that thereader can easily follow the conclusions.

• Data charts1: Each system configuration has a result data chart containing 6columns. Each row is an individual simulation. The first two columns specify thenumber of iterations2 performed. The third column indicates at which signal-to-noise ratio is the desired BER achieved3. The fourth column quantifies thenumber of nodes that have been extended in order to achieve this BER, which itwill be considered as a complexity factor. The next columns measure the relativecomplexity and SNR improvement with regard to the non-iterative case (casewith only 1 external iteration).

As this study is implementation independent4, the best or optimal trade-offs foreach system configuration will be considered the ones which achieve a BER of10−5 with the lowest Eb/N0, and they will marked with a darker backgroundin Table 3.2. Some simulations have data with more decimals because thoseconfigurations have been re-simulated with higher accuracy in the Matlab im-plementation.

• Graphic representations5: There are two graphics that combine the resultsof the 5 performed external iterations for each number of internal iterations:

– BER vs. Eb/N0: denoted as performance

– Nodes extended vs. Eb/N0: denoted as complexity

1Displayed in Appendix A.2int.it. = internal iterations (decoder iterations), ext.it = external iterations (detection ↔ decodingiterations)

3In this study, a BER of 10−5 has been used as the reference bit error rate to compare the results.Throughout this Chapter, this BER will be referred as our target error rate (TER).

4A specific latency value introduced by the iterations is not considered. However, if the best resultis achieved by two simulations, the one with a lower number of internal or external iterations isconsidered the optimal one. The latency influence will be discussed later in this Chapter.

5Displayed in Appendix B.

14

3.2. RESULTS

External Iterations Range of Complexity Increase

2 x2.1 - x2.43 x3.4 - x4.04 x4.7 - x5.6

Table 3.1: Detector complexity increase in iterative cases with regard to non-iterative cases

As it is uncomfortable for the reader to have all the tables and graphics togetherwith the results conclusions, they have been placed in the Appendixes A and B.

3.2 Results

When analysing the data charts and graphics, the results can be classified in severaltypes according to which feature of the system has influenced a concrete performanceor complexity degradation or increase. This is the way the case study will proceed. Asthe complete data charts are located in Appendix A and they are constantly referredin this chapter, a summary of all the result tables has been depicted in Table 3.2 foreasier consultation.

3.2.1 Iterations Configuration

It is substantial to make firstly an analysis based on the influence of the amount ofinternal and external iterations in the different system configurations, because furtherdiscussions will tackle the influence of other communications system model featuresrelated to the the amount of internal and external iterations.

To begin with, the sphere detector’s complexity (measured in the number of extendednodes) remains approximately constant independently of the number of decoder itera-tions. On the other hand, the BER performance improves with an increasing numberof FEC iterations because the decoder is able to correct more errors over the itera-tions.

An increasing number of external iterations increases the number of extended nodescompared to the non-iterative systems. Table 3.1 shows how this complexity fac-tor behaves for a different number of external iterations. It is observable that thecomplexity increases are within the same ranges for all the system configurationsanalysed.

15

3.2. RESULTS

12 14 16 18 20 22 24 260

10

20

30

40

50

60

Eb/N0 in dB

Aver

age

Num

ber o

f Ext

ende

d N

odes

5 ext. it.

1 ext. it.

(a) MIMO 2x2 64 QAM LDPC

10 15 20 250

50

100

150

200

250

300

350

400

450

Eb/N0 in dB

Aver

age

Num

ber o

f Ext

ende

d N

odes

5 ext it

1 ext it

(b) MIMO 4x4 64 QAM LDPC

Figure 3.1: Comparison of complexity of two system configurations

There is a particularity in the configurations with MIMO 2x2 64 QAM— The complex-ity representations in Appendix B show that the number of nodes extended is almostconstant, while in the remaining configurations is reduced, approximately linearly,with the Eb/N0 increase. See a comparison in Figure 3.1.

In an overview on Table 3.2 it is visible that the simulation results have only beendisplayed until four external iterations. A fifth detection↔ decoding iteration has beenomitted because the Eb/N0 improvement is not relevant6. Some other publicationshave already pointed out this iterative detection ↔ decoding performance threshold[YJ09] [tBKA04] over which no further significant signal-to-noise ratio savings can beachieved.

The maximum achievable SNR improvement, accomplished with 4 external iterations,is approximately of 3-4 dB in the case of 4x4 64 QAM, 3 dB in 4x4 16 QAM and 1-2 dBin 2x2 64 QAM (see Table 3.3). These improvements decrease slightly with an increas-ing number of internal iterations. For all configurations, and independently of the num-ber of decoder iterations, 2 external iterations provide the greatest SNR improvement(over the 70% of the maximum achievable SNR gain). Figure 3.2 depicts this behavior,which will be further analysed in the following sections taking into account the differentMIMO configurations, constellation sizes, and decoder types.

6In our case study, Eb/N0 variations under 0.1 dB are not relevant because the measurement errorsare higher.

16

3.2.RESU

LTS

int.it.(decoder)

ext.it.(det-dec)

Eb/N0 atBER≈10e-5

NodesExtended

Eb/N0 atBER≈10e-5

NodesExtended

Eb/N0 atBER≈10e-5

NodesExtended

Eb/N0 atBER≈10e-5

NodesExtended

Eb/N0 atBER≈10e-5

NodesExtended

Eb/N0 atBER≈10e-5

NodesExtended

1 1 ~ 19,4 dB 40 ~ 24,2 dB 42 ~ 19,2 dB 9 ~ 26,0 dB 9 ~ 13,9 dB 35 ~ 18,7 dB 37

1 2 ~ 16,8 dB 90 ~ 20,3 dB 87 ~ 17,4 dB 20 ~ 23,3 dB 20 ~ 11,6 dB 76 ~ 15,6 dB 75

1 3 ~ 16,0 dB 150 ~ 20,1 dB 135 ~ 17,1 dB 31 ~ 23,3 dB 30 ~ 11,0 dB 118 ~ 15,5 dB 114

1 4 ~ 15,6 dB 200 ~ 20,1 dB 180 ~ 17,1 dB 42 ~ 23,3 dB 40 ~ 10,8 dB 163 ~ 15,5 dB 153

2 1 ~ 17,8 dB 40 ~ 21,6 dB 40 ~ 17,3 dB 9 ~ 22,3 dB 9 ~ 12,3 dB 34 ~ 16,5 dB 35

2 2 ~ 15,2 dB 95 ~ 17,6 dB 90 ~ 15,7 dB 20 ~ 20,1 dB 20 ~ 10,0 dB 79 ~ 13,1 dB 75

2 3 ~ 14,4 dB 160 ~ 17,6 dB 140 ~ 15,7 dB 31 ~ 20,1 dB 31 ~ 9,5 dB 127 ~ 13,1 dB 114

2 4 ~ 14,2 dB 220 ~ 17,6 dB 190 ~ 15,6 dB 42 ~ 20,1 dB 41 ~ 9,4 dB 173 ~ 13,1 dB 153

3 1 ~ 17,5 dB 40 ~ 20,0 dB 40 ~ 17,0 dB 9 ~ 20,0 dB 9 ~ 12,0 dB 34 ~ 15,0 dB 35

3 2 ~ 15,2 dB 100 ~ 16,1 dB 95 ~ 15,4 dB 20 ~ 18,0 dB 20 ~ 9,8 dB 80 ~ 11,7 dB 75

3 3 ~ 14,3 dB 160 ~ 16,0 dB 145 ~ 15,5 dB 31 ~ 18,0 dB 31 ~ 9,5 dB 125 ~ 11,7 dB 117

3 4 ~ 14,1 dB 225 ~ 16,0 dB 200 ~ 15,3 dB 42 ~ 18,0 dB 42 ~ 9,2 dB 177 ~ 11,7 dB 158

4 1 ~ 17,45 dB 40 ~ 19,0 dB 40 ~ 16,9 dB 9 ~ 18,6 dB 9 ~ 12,0 dB 34 ~ 14,0 dB 35

4 2 ~ 15,13 dB 95 ~ 15,4 dB 95 ~ 15,3 dB 20 ~ 17,0 dB 20 ~ 9,8 dB 80 ~ 11,0 dB 75

4 3 ~ 14,33 dB 160 ~ 15,3 dB 150 ~ 15,3 dB 31 ~ 17,0 dB 31 ~ 9,4 dB 128 ~ 10,9 dB 118

4 4 ~ 13,98 dB 222 ~ 15,3 dB 200 ~ 15,3 dB 43 ~ 17,0 dB 43 ~ 9,2 dB 179 ~ 10,9 dB 161

5 1 ~ 17,40 dB 40 ~ 18,3 dB 40 ~ 16,8 dB 9 ~ 17,8 dB 9 ~ 12,0 dB 34 ~ 13,4 dB 36

5 2 ~ 15,16 dB 96 ~ 15,0 dB 95 ~ 15,3 dB 20 ~ 16,4 dB 20 ~ 9,8 dB 79 ~ 10,5 dB 76

5 3 ~ 14,30 dB 164 ~ 14,8 dB 155 ~ 15,3 dB 31 ~ 16,4 dB 32 ~ 9,4 dB 128 ~ 10,5 dB 120

5 4 ~ 13,91 dB 224 ~ 14,8 dB 210 ~ 15,2 dB 42 ~ 16,4 dB 43 ~ 9,2 dB 178 ~ 10,5 dB 164

6 1 ~ 17,28 dB 39 ~ 17,9 dB 40 ~ 16,7 dB 9 ~ 17,3 dB 9 ~ 11,8 dB 34 ~ 13,0 dB 34

6 2 ~ 15,06 dB 97 ~ 14,74 dB 99 ~ 15,2 dB 20 ~ 16,2 dB 20 ~ 9,8 dB 79 ~ 10,4 dB 77

6 3 ~ 14,29 dB 162 ~ 14,64 dB 156 ~ 15,3 dB 31 ~ 16,2 dB 32 ~ 9,3 dB 131 ~ 10,2 dB 122

6 4 ~ 13,88 dB 227 ~ 14,62 dB 214 ~ 15,2 dB 42 ~ 16,2 dB 43 ~ 9,2 dB 179 ~ 10,2 dB 166

7 1 ~ 17,3 dB 40 ~ 17,6 dB 40 ~ 16,7 dB 9 ~ 17,0 dB 9 ~ 11,8 dB 34 ~ 12,7 dB 34

7 2 ~ 15,1 dB 95 ~ 14,65 dB 98 ~ 15,1 dB 20 ~ 16,1 dB 20 ~ 9,8 dB 79 ~ 10,3 dB 77

7 3 ~ 14,2 dB 160 ~ 14,53 dB 154 ~ 15,3 dB 31 ~ 16,1 dB 32 ~ 9,3 dB 131 ~ 10,2 dB 121

7 4 ~ 14,1 dB 220 ~ 14,51 dB 212 ~ 15,2 dB 42 ~ 16,1 dB 43 ~ 9,2 dB 179 ~ 10,0 dB 167

8 1 ~ 17,3 dB 40 ~ 17,4 dB 40 ~ 16,5 dB 9 ~ 16,7 dB 9 ~ 11,9 dB 34 ~ 12,5 dB 34

8 2 ~ 15,0 dB 95 ~ 14,58 dB 98 ~ 15,1 dB 20 ~ 16,1 dB 20 ~ 9,7 dB 80 ~ 10,2 dB 77

8 3 ~ 14,3 dB 165 ~ 14,52 dB 155 ~ 15,2 dB 31 ~ 16,0 dB 32 ~ 9,3 dB 131 ~ 10,1 dB 121

8 4 ~ 14,0 dB 220 ~ 14,48 dB 213 ~ 15,3 dB 42 ~ 16,1 dB 43 ~ 9,2 dB 180 ~ 10,1 dB 166

9 1 ~ 17,2 dB 40 ~ 17,3 dB 40 ~ 16,6 dB 9 ~ 16,5 dB 9 ~ 11,8 dB 34 ~ 12,3 dB 34

9 2 ~ 15,1 dB 95 ~ 14,57 dB 97 ~ 15,2 dB 20 ~ 16,1 dB 20 ~ 9,7 dB 80 ~ 10,1 dB 77

9 3 ~ 14,2 dB 165 ~ 14,51 dB 157 ~ 15,1 dB 31 ~ 16,0 dB 32 ~ 9,3 dB 132 ~ 10,1 dB 121

9 4 ~ 14,0 dB 225 ~ 14,43 dB 212 ~ 15,1 dB 42 ~ 16,0 dB 43 ~ 9,3 dB 179 ~ 10,1 dB 166

10 1 ~ 17,2 dB 40 ~ 17,2 dB 40 ~ 16,5 dB 9 ~ 16,4 dB 9 ~ 11,8 dB 34 ~ 12,2 dB 34

10 2 ~ 15,0 dB 100 ~ 14,5 dB 100 ~ 15,1 dB 20 ~ 16,1 dB 20 ~ 9,7 dB 80 ~ 10,1 dB 78

10 3 ~ 14,2 dB 165 ~ 14,5 dB 155 ~ 15,2 dB 31 ~ 16,0 dB 32 ~ 9,3 dB 131 ~ 10,1 dB 122

10 4 ~ 14,0 dB 220 ~ 14,5 dB 210 ~ 15,1 dB 42 ~ 16,0 dB 43 ~ 9,1 dB 182 ~ 10,0 dB 165

4x4 16QAM PCCC 4x4 16QAM LDPCCONFIGURATION 4x4 64QAM PCCC 4x4 64QAM LDPC 2x2 64QAM PCCC 2x2 64QAM LDPC

Table 3.2: Summary of all the data charts in Appendix A. In grey the configurations with best performance-complexity trade-offs

17

3.2. RESULTS

Configuration SNR Improvement

4x4 64 QAM 3-4 dB4x4 16 QAM 3 dB2x2 64 QAM 1-2 dB

Table 3.3: Maximum achievable SNR improvement, accomplished with 4 externaliterations, for different system configurations

As introduced in section 3.1, specific latency values (in clock cycles) are not as-sumed, therefore only the number of iterations is taken as a measure of latency.However, it should be mentioned that, when taking into account a concrete im-plementation, both the detector and the detection ↔ decoding iterations incur la-tency:

• Sphere Detector: Its latency is measured as the number of extended nodes andit depends on how many cycles our implementation needs to extend a node.

Sphere Detector latency = number of nodes ∗ number of clock cycles per node(3.1)

• FEC latency: It depends on the implementation and the number of iterations.

FEC latency = iteration latency ∗ number of iterations (3.2)

Therefore, choosing configurations with a high number of internal or external iterationsover those with fewer number of iterations, when only a small performance gain isachieved, would depend on the accuracy needed and on the computational resourcesat hand in a concrete implementation. However, increasing the number of internaliterations is generally preferable rather than increasing the external ones, since latencycost of external iterations involves the increased latency of repeated detection anddecoding stages.

18

3.2.RESU

LTS

16,7

16,1 16,0 16,1

17,3

15,0

14,3 14,0

17,4

14,6 14,5

14,5

16,5

15,1 15,2 15,3

12,5

10,2 10,1 10,1

11,9

9,7 9,3 9,2

9,0

10,0

11,0

12,0

13,0

14,0

15,0

16,0

17,0

18,0

1 2 3 4

E b/N

0 (dB

)

External iterations (detection-decoding)

2x2 64 QAM PCCC 2x2 64 QAM LDPC



64 QAM 16 QAM

MIMO 2x2 MIMO 4x4

PCCC LDPC

Figure 3.2: Eb/N0 at a BER of 10−5, with 8 internal iterations, for a different number of external iterations, for different systemconfigurations

19

3.2. RESULTS

PCCC LDPCExternal Iterations Internal Iterations Range Internal Iterations Range

1 6 82 4 - 5 7 - 83 4 - 5 74 3 - 5 7

Table 3.4: Relationship between the number of external and internal iterations inan optimal case

3.2.1.1 Internal-External Iterations Relationship

Throughout this case study, it has been found an approximate relationship betweenthe number of external and internal iterations needed to achieve the optimal con-figuration7 in the different system configurations. In Figure 3.3 this dependency isdepicted.

For a given number of detection↔ decoding iterations it is predictable an approximateoptimal number of internal iterations (decoder) to achieve a BER of 10−5 at the lowestEb/N0 possible. Although the relationship is not exact because it varies with thesystem configuration, there are some iteration ranges where choosing a number ofiterations would ensure a quasi-optimal configuration (see Table 3.4). In general, theoptimal internal iterations range for PCCC is between 3 and 6, while for LDPC is 7or 8.

Taking Table 3.4 into consideration, a quasi-optimal number of internal iterations canbe chosen given the number of external iterations. The election of a concrete numberof iterations within the confidence intervals given in the Table 3.4 would depend onthe specifications of the implementation.

3.2.2 MIMO Configuration

In Table 3.2 it is observable that MIMO configurations with a larger number of anten-nas affect the sphere detector’s complexity, what has already been quantified in otherpublications like [AMF11]. This behavior will be discussed next.

7The optimal configurations used to create Figure 3.3 may differ (one iteration up or down) depend-ing on the application requirements (greater importance of latency or SNR) as well as due to thesimulations precision error.

20

3.2. RESULTS

2

3

4

5

6

7

8

9

10

1 2 3 4

Inte

rnal

iter

atio

ns (d

ecod

er)

External iterations (detection-decoding)

2x2 64QAM PCCC 4x4 64QAM PCCC 4x4 16QAM PCCC

2x2 64QAM LDPC 4x4 64QAM LDPC 4x4 16QAM LDPC

LDPC

PCCC

Figure 3.3: Relationship between internal and external iterations for the optimalconfigurations (lowest SNR at BER=10−5) in different system config-urations

2x2 64 QAM PCCC vs. 4x4 64 QAM PCCC It is observable that the 2x2 systemperforms always better than the 4x4 system when using the non-iterative system(i.e. 1 external iteration). The SNR improvement ranges from 0.2 dB to 0.7 dB,increasing with a higher number of internal iterations (see Table 3.5). In [AMF11],using 8 internal iterations, the same comparison yielded an SNR improvement ofapproximately 2 dB, which is a bit better than in our case study because bigger codeblocks and a random interleaver are used.

Nonetheless, when the number of external iterations increases, the 4x4 system outper-forms the SNR performance of the 2x2 system, as observed in Figure 3.2, especially

Internal Iterations Relative SNR Improvement

1 0.2 dB2 - 4 0.5 dB5 - 7 0.6 dB8 - 10 0.7 dB

Table 3.5: Relative SNR improvement between MIMO 2x2 64 QAM PCCC and4x4 64 QAM PCCC performing 1 external iteration for a different num-ber of internal iterations

21

3.2. RESULTS

Internal Iterations Relative SNR Improvement

4 0.4 dB5 0.5 dB

6 - 7 0.6 dB8 0.7 dB

9 - 10 0.8 dB

Table 3.6: Relative SNR improvement between MIMO 2x2 64 QAM LDPC and4x4 64 QAM LDPC performing 1 external iteration for a different num-ber of internal iterations

when it has less than 6 internal iterations. For 2 external iterations and more than7 internal iterations the performance of the 2x2 and 4x4 is very similar. Reason forthis is that for more than 2 external iterations the 4x4 system still provides someSNR improvement while the 2x2 does not. Additionally, it should be noticed that theclipping value was near optimal for 4x4. SNR performance of the 2x2 system could beslightly improved by adjusting the clipping value accordingly.

The reader can notice that for 3 and 4 external iterations, the 2x2 systems perform abit worse (0.1 dB) than with less external iterations. This is an effect of the simulationprecision.

2x2 64 QAM LDPC vs. 4x4 64 QAM LDPC In case LDPC is employed, the resultsare similar but with some particularities. 2x2 performs better than 4x4 when the non-iterative system is used, except when 3 or less internal iterations are performed. Inthis case, 2x2 performs 0-2 dB worse than 4x4. For the remaining cases it happens asfor PCCC: The SNR improvement ranges 0.4 dB - 0.8 dB, increasing with a highernumber of internal iterations (see Table 3.6).

When the number of external iterations increases, the 4x4 system outperforms theSNR performance of the 2x2 system. The abovementioned behaviors of the 2x2 and4x4 systems for a different number of external iterations vs. SNR are noticeable inFigure 3.2.

The same conclusion can be extracted from the two previous comparisons: The benefitof iterative detection↔ decoding is not so significative in systems with a small numberof antennas and big constellation sizes, and this effect is even worse when LDPC is usedrather than PCCC. The only advantage of these systems is that latency of the spheredetector would be smaller. Unless the latency were a critical application parameter,using MIMO 4x4 instead of MIMO 2x2 would provide a better spectral efficiency aswell as BER performance.

22

3.2. RESULTS

FEC it.(int.it)

SD-FEC it.(ext.it)

Eb/N0 atBER≈10e-5

SNRImprovement

Eb/N0 atBER≈10e-5

SNRImprovement

Eb/N0 atBER≈10e-5

SNRImprovement

Eb/N0 atBER≈10e-5

SNRImprovement

1 4 ~ 15,6 dB ~ 3,8 dB ~ 10,8 dB ~ 3,1 dB ~ 20,1 dB ~ 4,1 dB ~ 15,5 dB ~ 3,2 dB2 4 ~ 14,2 dB ~ 3,6 dB ~ 9,4 dB ~ 2,9 dB ~ 17,6 dB ~ 4,0 dB ~ 13,1 dB ~ 3,4 dB3 4 ~ 14,1 dB ~ 3,4 dB ~ 9,2 dB ~ 2,8 dB ~ 16,0 dB ~ 4,0 dB ~ 11,7 dB ~ 3,3 dB4 4 ~ 13,98 dB ~ 3,47 dB ~ 9,2 dB ~ 2,8 dB ~ 15,3 dB ~ 3,7 dB ~ 10,9 dB ~ 3,1 dB5 4 ~ 13,91 dB ~ 3,49 dB ~ 9,2 dB ~ 2,8 dB ~ 14,8 dB ~ 3,5 dB ~ 10,5 dB ~ 2,9 dB6 4 ~ 13,88 dB ~ 3,40 dB ~ 9,2 dB ~ 2,6 dB ~ 14,62 dB ~ 3,28 dB ~ 10,2 dB ~ 2,8 dB7 4 ~ 14,1 dB ~ 3,2 dB ~ 9,2 dB ~ 2,6 dB ~ 14,51 dB ~ 3,09 dB ~ 10,0 dB ~ 2,7 dB8 4 ~ 14,0 dB ~ 3,3 dB ~ 9,2 dB ~ 2,7 dB ~ 14,48 dB ~ 2,92 dB ~ 10,1 dB ~ 2,4 dB9 4 ~ 14,0 dB ~ 3,2 dB ~ 9,3 dB ~ 2,5 dB ~ 14,43 dB ~ 2,87 dB ~ 10,1 dB ~ 2,2 dB10 4 ~ 14,0 dB ~ 3,2 dB ~ 9,1 dB ~ 2,7 dB ~ 14,5 dB ~ 2,7 dB ~ 10,0 dB ~ 2,2 dB

4x4 64 QAM PCCC 4x4 16 QAM PCCC 4x4 64 QAM LDPC 4x4 16 QAM LDPC

Table 3.7: SNR improvement with 4 external iterations compared with only 1 ex-ternal iteration, for 64 QAM and 16 QAM systems

Configuration SNR Improvement Range (best - worst)

4x4 64 QAM PCCC 3.8 - 3.2 dB4x4 64 QAM LDPC 4.1 - 2.7 dB4x4 16 QAM PCCC 3.1 - 2.5 dB4x4 16 QAM LDPC 3.2 - 2.2 dB

Table 3.8: Summary of Table 3.7

3.2.3 Constellation Size

When it comes to the constellation size, since the computational complexity at thesphere detector is increased when the constellation size is bigger [AMF11] [BZF06],it is expectable for smaller constellation sizes that the number of extended nodeswould be reduced. In this section has been studied how the number of extendednodes changes in two different constellation sizes for different iterations configura-tions.

The proportion between the number of extended nodes in 4x4 64 QAM and in 4x4 16 QAMremains unaffected by the number of both the external and internal iterations. The16 QAM system has always an approximately 90% of the number of nodes extendedby the 64 QAM system for 1 external iteration, and 80% for 2 to 4 external itera-tions. The percentage does not vary with the number of internal iterations, and canbe applied to both the PCCC and the LDPC case.

Moreover, the SNR gain in iterative systems is smaller for "smaller systems" (16 QAMin comparison with 64 QAM). While in MIMO 4x4 64 QAM PCCC the difference(at 4 internal iterations) between using 1 and 4 external iterations is approximately3.5 dB, this difference is only of 2.8 dB in MIMO 4x4 16 QAM PCCC. It also de-creases slightly with an increasing number of internal iterations (see Tables 3.7 and3.8).

23

3.2. RESULTS

In Table 3.9 it is noticeable that the relative SNR gain between the SNR improvementof iterative systems (column "A-B") between 64 QAM and 16 QAM systems is alwaysbetween 0.1 dB and 0.7 dB (higher values for higher number of external iterations)using PCCC, and almost constantly 0.6 dB using LDPC.

24

3.2.RESU

LTS

A B C DFEC it.(int.it)

SD-FEC it.(ext.it)

Eb/N0 atBER≈10e-5

SNRImprovement

Eb/N0 atBER≈10e-5

SNRImprovement

A - BEb/N0 at

BER≈10e-5SNR

ImprovementEb/N0 at

BER≈10e-5SNR

ImprovementC - D

1 1 ~ 19,4 dB ~ 13,9 dB ~ 24,2 dB ~ 18,7 dB1 2 ~ 16,8 dB ~ 2,6 dB ~ 11,6 dB ~ 2,3 dB ~ 0,3 dB ~ 20,3 dB ~ 3,9 dB ~ 15,6 dB ~ 3,1 dB ~ 0,8 dB1 3 ~ 16,0 dB ~ 3,4 dB ~ 11,0 dB ~ 2,9 dB ~ 0,5 dB ~ 20,1 dB ~ 4,1 dB ~ 15,5 dB ~ 3,2 dB ~ 0,9 dB1 4 ~ 15,6 dB ~ 3,8 dB ~ 10,8 dB ~ 3,1 dB ~ 0,7 dB ~ 20,1 dB ~ 4,1 dB ~ 15,5 dB ~ 3,2 dB ~ 0,9 dB2 1 ~ 17,8 dB ~ 12,3 dB ~ 21,6 dB ~ 16,5 dB2 2 ~ 15,2 dB ~ 2,6 dB ~ 10,0 dB ~ 2,3 dB ~ 0,3 dB ~ 17,6 dB ~ 4,0 dB ~ 13,1 dB ~ 3,4 dB ~ 0,6 dB2 3 ~ 14,4 dB ~ 3,4 dB ~ 9,5 dB ~ 2,8 dB ~ 0,6 dB ~ 17,6 dB ~ 4,0 dB ~ 13,1 dB ~ 3,4 dB ~ 0,6 dB2 4 ~ 14,2 dB ~ 3,6 dB ~ 9,4 dB ~ 2,9 dB ~ 0,7 dB ~ 17,6 dB ~ 4,0 dB ~ 13,1 dB ~ 3,4 dB ~ 0,6 dB3 1 ~ 17,5 dB ~ 12,0 dB ~ 20,0 dB ~ 15,0 dB3 2 ~ 15,2 dB ~ 2,3 dB ~ 9,8 dB ~ 2,2 dB ~ 0,1 dB ~ 16,1 dB ~ 3,9 dB ~ 11,7 dB ~ 3,3 dB ~ 0,6 dB3 3 ~ 14,3 dB ~ 3,2 dB ~ 9,5 dB ~ 2,5 dB ~ 0,7 dB ~ 16,0 dB ~ 4,0 dB ~ 11,7 dB ~ 3,3 dB ~ 0,7 dB3 4 ~ 14,1 dB ~ 3,4 dB ~ 9,2 dB ~ 2,8 dB ~ 0,6 dB ~ 16,0 dB ~ 4,0 dB ~ 11,7 dB ~ 3,3 dB ~ 0,7 dB4 1 ~ 17,45 dB ~ 12,0 dB ~ 19,0 dB ~ 14,0 dB4 2 ~ 15,13 dB ~ 2,32 dB ~ 9,8 dB ~ 2,2 dB ~ 0,1 dB ~ 15,4 dB ~ 3,6 dB ~ 11,0 dB ~ 3,0 dB ~ 0,6 dB4 3 ~ 14,33 dB ~ 3,12 dB ~ 9,4 dB ~ 2,6 dB ~ 0,5 dB ~ 15,3 dB ~ 3,7 dB ~ 10,9 dB ~ 3,1 dB ~ 0,6 dB4 4 ~ 13,98 dB ~ 3,47 dB ~ 9,2 dB ~ 2,8 dB ~ 0,7 dB ~ 15,3 dB ~ 3,7 dB ~ 10,9 dB ~ 3,1 dB ~ 0,6 dB5 1 ~ 17,40 dB ~ 12,0 dB ~ 18,3 dB ~ 0,7 dB ~ 13,4 dB5 2 ~ 15,16 dB ~ 2,24 dB ~ 9,8 dB ~ 2,2 dB ~ 0,0 dB ~ 15,0 dB ~ 3,3 dB ~ 10,5 dB ~ 2,9 dB ~ 0,4 dB5 3 ~ 14,30 dB ~ 3,10 dB ~ 9,4 dB ~ 2,6 dB ~ 0,5 dB ~ 14,8 dB ~ 3,5 dB ~ 10,5 dB ~ 2,9 dB ~ 0,6 dB5 4 ~ 13,91 dB ~ 3,49 dB ~ 9,2 dB ~ 2,8 dB ~ 0,7 dB ~ 14,8 dB ~ 3,5 dB ~ 10,5 dB ~ 2,9 dB ~ 0,6 dB6 1 ~ 17,28 dB ~ 11,8 dB ~ 17,9 dB ~ 13,0 dB6 2 ~ 15,06 dB ~ 2,22 dB ~ 9,8 dB ~ 2,0 dB ~ 0,2 dB ~ 14,74 dB ~ 3,16 dB ~ 10,4 dB ~ 2,6 dB ~ 0,6 dB6 3 ~ 14,29 dB ~ 2,99 dB ~ 9,3 dB ~ 2,5 dB ~ 0,5 dB ~ 14,64 dB ~ 3,26 dB ~ 10,2 dB ~ 2,8 dB ~ 0,5 dB6 4 ~ 13,88 dB ~ 3,40 dB ~ 9,2 dB ~ 2,6 dB ~ 0,8 dB ~ 14,62 dB ~ 3,28 dB ~ 10,2 dB ~ 2,8 dB ~ 0,5 dB7 1 ~ 17,3 dB ~ 11,8 dB ~ 17,6 dB ~ 12,7 dB7 2 ~ 15,1 dB ~ 2,2 dB ~ 9,8 dB ~ 2,0 dB ~ 0,2 dB ~ 14,65 dB ~ 2,95 dB ~ 10,3 dB ~ 2,4 dB ~ 0,6 dB7 3 ~ 14,2 dB ~ 3,1 dB ~ 9,3 dB ~ 2,5 dB ~ 0,6 dB ~ 14,53 dB ~ 3,07 dB ~ 10,2 dB ~ 2,5 dB ~ 0,6 dB7 4 ~ 14,1 dB ~ 3,2 dB ~ 9,2 dB ~ 2,6 dB ~ 0,6 dB ~ 14,51 dB ~ 3,09 dB ~ 10,0 dB ~ 2,7 dB ~ 0,4 dB8 1 ~ 17,3 dB ~ 11,9 dB ~ 17,4 dB ~ 12,5 dB8 2 ~ 15,0 dB ~ 2,3 dB ~ 9,7 dB ~ 2,2 dB ~ 0,1 dB ~ 14,58 dB ~ 2,82 dB ~ 10,2 dB ~ 2,3 dB ~ 0,5 dB8 3 ~ 14,3 dB ~ 3,0 dB ~ 9,3 dB ~ 2,6 dB ~ 0,4 dB ~ 14,52 dB ~ 2,88 dB ~ 10,1 dB ~ 2,4 dB ~ 0,5 dB8 4 ~ 14,0 dB ~ 3,3 dB ~ 9,2 dB ~ 2,7 dB ~ 0,6 dB ~ 14,48 dB ~ 2,92 dB ~ 10,1 dB ~ 2,4 dB ~ 0,5 dB9 1 ~ 17,2 dB ~ 11,8 dB ~ 17,3 dB ~ 12,3 dB9 2 ~ 15,1 dB ~ 2,1 dB ~ 9,7 dB ~ 2,1 dB ~ 0,0 dB ~ 14,57 dB ~ 2,73 dB ~ 10,1 dB ~ 2,2 dB ~ 0,5 dB9 3 ~ 14,2 dB ~ 3,0 dB ~ 9,3 dB ~ 2,5 dB ~ 0,5 dB ~ 14,51 dB ~ 2,79 dB ~ 10,1 dB ~ 2,2 dB ~ 0,6 dB9 4 ~ 14,0 dB ~ 3,2 dB ~ 9,3 dB ~ 2,5 dB ~ 0,7 dB ~ 14,43 dB ~ 2,87 dB ~ 10,1 dB ~ 2,2 dB ~ 0,7 dB10 1 ~ 17,2 dB ~ 11,8 dB ~ 17,2 dB ~ 12,2 dB10 2 ~ 15,0 dB ~ 2,2 dB ~ 9,7 dB ~ 2,1 dB ~ 0,1 dB ~ 14,5 dB ~ 2,7 dB ~ 10,1 dB ~ 2,1 dB ~ 0,6 dB10 3 ~ 14,2 dB ~ 3,0 dB ~ 9,3 dB ~ 2,5 dB ~ 0,5 dB ~ 14,5 dB ~ 2,7 dB ~ 10,1 dB ~ 2,1 dB ~ 0,6 dB10 4 ~ 14,0 dB ~ 3,2 dB ~ 9,1 dB ~ 2,7 dB ~ 0,5 dB ~ 14,5 dB ~ 2,7 dB ~ 10,0 dB ~ 2,2 dB ~ 0,5 dB

4x4 64 QAM PCCC 4x4 16 QAM PCCC 4x4 64 QAM LDPC 4x4 16 QAM LDPC

Table 3.9: Relative SNR gain between the SNR iterative improvement (gain of iterative configurations compared with the non-iterativeones) between 4x4 16 QAM and 4x4 64 QAM systems, using PCCC and LDPC

25

3.2. RESULTS

LDPC PCCC

External Iterations SNR Gain Complexity Increase SNR Gain Complexity Increase

2 90%-100% x2.3 70% x2.33 - - 90% x4

Table 3.10: SNR gain and complexity increase in iterative cases with regard tonon-iterative cases, for different decoders

3.2.4 Decoder Type

The chosen FEC strategy has also an impact on the performance-complexity trade-offs.For all configurations, using LDPC and independently of the number of FEC iterations,2 external iterations achieve already 90-100% of the maximum SNR improvement (4external iterations) at a complexity of approximately x2.3 with regard to the the non-iterative case (see Table 3.10). Therefore, with LDPC there is no benefit in performingmore than 2 external iterations. In the case of PCCC, 2 external iterations achievearound 70% of the maximum SNR improvement at a complexity of approximatelyx2.3 with regard to the non-iterative case, and 3 external iterations achieve around90% of the maximum SNR improvement at a complexity of approximately x4 withregard to the non-iterative case. Therefore, with PCCC it is worth performing up to 3external iterations, or even 4 in those cases where the BER performance is much moreimportant than complexity. Despite this behavior, the overall performance of LDPCis worse because requires a higher number of internal iterations than PCCC for thesame BER performance (see Figure 3.3).

It is noticeable that the difference in the number of nodes extended, between twoconfigurations where only the FEC varies, is higher with an increasing number ofinternal and external iterations.

Additionally, the difference between the number of nodes extended with LDPC vs.PCCC is slightly growing with smaller constellation sizes. Let us explain this behav-ior with an example. MIMO 4x4 64 QAM PCCC and MIMO 4x4 64 QAM LDPC: At10 internal iterations and 4 external iterations using PCCC, 220 nodes are extended,and using LDPC only 210 are extended. For a smaller constellation size, this reductionin the number of extended nodes is even bigger, having a difference of approximately 20extended nodes between MIMO 4x4 16 QAM PCCC and MIMO 4x4 16 QAM LDPCat 10 internal iterations and 4 external iterations (182 and 165 nodes extended respec-tively).

26

3.2. RESULTS

PCCC LDPCSystem Configuration Extended Nodes Extended Nodes

MIMO 4x4 64 QAM 220 210MIMO 4x4 16 QAM 182 165MIMO 2x2 64 QAM 42 43

Table 3.11: Number of extended nodes for different system configurations, at 10internal iterations and 4 external iterations

Table 3.11 shows how the number of extended nodes is reduced when combining the in-fluence of the decoder, the constellation size, and the MIMO configuration.

The configurations using LDPC extend generaly less nodes. There is an exception: Thenumber of extended nodes is not significantly affected between MIMO 2x2 64 QAMPCCCand MIMO 2x2 64 QAM LDPC, which indicates that in smaller MIMO configurationswith big constellation sizes the difference in complexity given by the election of thedecoder is not relevant.

3.2.5 Interleaver Type

Modifications concerning the interleaver type have been considered for additional com-parison. Figures 3.4(a) and 3.4(b) represent the performance and complexity analysis,respectively, of a MIMO 4x4 64 QAM PCCC system with 5 internal and 4 externaliterations8, changing the block interleaver to a random interleaver. In both cases thelength of the interleavers is the same.

The reader can see that this modification in the simulation setup yields an Eb/N0

gain at a BER=10−5 of around 1.5 dB. The BER decreases much faster with therandom interleaver than with the block interleaver. The complexity when using therandom interleaver is much higher for lower SNR scenarios than with block interleaver,nevertheless it increases very fast with the random interleaver and at BER=10−5 thenumber of extended nodes is very similar, being ≈ 240 for the random interleaver and≈ 220 for the block interleaver.

8Which is within the range of quasi-optimal configurations. See Table 3.4.

27

3.2. RESULTS

8 9 10 11 12 13 14 1510 6

10 5

10 4

10 3

10 2

10 1

100

BER

Eb/N0 in dB

Block InterleaverRandom Interleaver

(a) Performance

8 9 10 11 12 13 14 15200

250

300

350

400

450

500

550

600

650

Eb/N0 in dB

Aver

age

Num

ber o

f Ext

ende

d N

odes

Block InterleaverRandom Interleaver

(b) Complexity

Figure 3.4: Comparison of performance and complexity for different interleavertypes and block sizes for a MIMO 4x4 64 QAM PCCC system

28

3.3. SUMMARY

3.3 Summary

In general, the number of extended nodes is not significantly increased with inter-nal iterations but with external iterations (see Table 3.1), because the latest are theones which involve the detector, and thus the tree search that produces the nodesextension.

Normally, increasing the number of internal iterations means increasing the BER per-formance, but there are a certain number of internal iterations beyond which thereis no significant performance improvement. 4-6 internal iterations for PCCC and 7-8 for LDPC provide good performance-complexity trade-offs. The same applies forthe detection ↔ decoding iterations, where 70-90% of the the maximum achievablerelative SNR gain over non-iterative systems is achieved with 2 external iterations. Ta-ble 3.2 sumarises the trade-offs between the number of internal iterations and externaliterations to achieve the best performance.

Therefore, a certain BER performance can be achieved with more than one config-uration (different number of antennas, constellation size, decoder type, internal andexternal iterations), and the best trade-off between complexity and communicationsperformance depends on the application scenario. In general, minimizing the numberof external iterations is a priority, since they are the ones which incur more latency(latency of the FEC plus latency of the SD).

In addition to the conclusions obtained from the study cases, the data charts andgraphic representations from all the system configurations constitute a valuable sourceof information for comparability and reference with other publications and futurework.

29

4 Adaptive LLR Clipping

4.1 Motivation

It has been demonstrated that the use of an adequate LLR clipping value is crucialfor good performance in the detector [dJW05] [MFF09]. From (2.4) can be inferredthat a higher |Le(u)| provides a better (lower) BER. That is logical, because a higherLLR ratio indicates higher reliability in the decision of choosing 1 or 0. Taking theprevious into consideration, it can be stated that the adjustment of the BER is plausi-ble by clipping the L-values. As the BER can be estimated by means of the L-values,controlling them can lead to the optimization of the transmission system to fit ourTER (Target Error Rate).

In the previous chapter, where several system configurations were analysed, all config-urations used the same the LLR clipping value Lcl. An Lcl of 4.4, taken from previouswork [MvBF09], was used along all the study cases. Let us start with a comparison(Figure 4.1) between two system configurations where the only change was the con-stellation size (16 QAM and 64 QAM). Without knowing which is in each case theneeded Lcl to fit our TER, it is expectable that for 16 QAM the needed Lcl may behigher, because in small systems, as well as in systems with big tuple sizes T , mostcounter-hypotheses are found, so clipping of L-values does not need to be so aggres-sive. The main objective in this case is just avoid them to be infinite. For greatersystems as well as systems with smaller T , only a few counter-hypotheses are found,and good performance is only achieved with a more aggressive clipping, which avoidsoverestimating the reliability of the non-found counter-hypotheses. The objective inthis case is not just to avoid the L-values to be infinite, but also to avoid overestimatethem.

As expected, a difference is found between the SNR needed to achieve a BER of 10−5

in the 16 QAM and the 64 QAM case. This difference would vary slightly if theadequate clipping value were used in each case.

In [MvBF09] a method is used to find the optimal LLR clipping value for a specifictransmission configuration. It consists of analysing, for a given TER and the corre-

31

4.1. MOTIVATION

6 8 10 12 14 1610 6

10 5

10 4

10 3

10 2

10 1

100

BER

Eb/N0 in dB

1 ext it5 ext it

(a) 16 QAM

10 12 14 16 18 20 2210 6

10 5

10 4

10 3

10 2

10 1

100

BER

Eb/N0 in dB

5 ext. it. 1 ext. it.

(b) 64 QAM

Figure 4.1: Analysis of BER vs. Complexity of two transmission systems withQAM modulations, MIMO 4x4, Sphere Detector, PCCC Decoder, 1internal iteration, 1-5 external iterations, clipping value=4.4

4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5 5.6 5.70.5743

0.5744

0.5745

0.5746

0.5747

0.5748

0.5749

0.575

0.5751

LLR Clipping Value

Mut

ual I

nfor

mat

ion

Figure 4.2: Mutual Information

sponding BER, the mutual information between the binary input and LLR output ofthe MIMO detector (see Figure 4.2). The optimal LLR clipping value for a (SNR,TER) tuple is the one for which the average mutual information at the detector’soutput is maximized. That would be to choose the clipping value corresponding tothe maximum of the mutual information curve in the Figure 4.2.

Although this approach provides an accurate clipping value for a given SNR and TER,a performance prediction burden arises because the mutual information calculationrequires extensive simulations.

An alternative to the mutual information approach is proposed in [NA11], where theLLR clipping value is determined adaptively. Through this approach the Lcl that

32

4.2. ALGORITHM AND IMPLEMENTATION

Binary Data Source Outer Encoder Interleaver

∏Constellation

Mapper

Binary Data Sink Outer Decoder

Deinterleaver∏-1

Detector /Demapper

Adaptive Clipping

Interleaver∏

+

+

External Clipping

Transmitter

Receiver

QAM

û

u c' c

Le(u) La(c')

Le(c') La(c)

Le(c)

Internal Clipping (Radius)

-

NR

NTi.i.d. Rayleigh flat channel

La,cl(c')

Lcl

L(c)

-

Figure 4.3: Comunications System Model with Adaptive LLR Clipping

fits our TER is not calculated "off-line" (mutual information), but on-the-fly. Thisapproach will be tackled in the following section.

4.2 Algorithm and Implementation

There are different vectors of L-values within our communications system model thatcontribute to the BER of the transmission system. Although the L-values at thedecoder’s output (Le(u)) are the ones that contribute directly to the final BER asexpressed in (2.4), they are a result of the input to the decoder, which is the de-tector’s deinterleaved L-values vector (La(c’)). As the BER is expected to be bet-ter after channel decoding due to the employment of FEC codes, in [NA11] is sug-gested that the clipping of the LLR values should be done just before channel decod-ing.

In order to implement the adaptive LLR clipping algorithm proposed in [NA11], inthis study the communications system model from Figure 2.1 has been modified to addthe blocks that could perform the adaptive clipping: Adaptive Clipping and ExternalClipping blocks. The new model can be found in Figure 4.3. An External Clippingblock already existed in our fundamental communications system model, but it wasimplicit in the Outer Decoder block.

The method proposed in [NA11] is described step by step as follows:

1. LLR clipping value initialization

2. Estimation of the previous frame BER

33


3. Comparison of the current frame BER with the TER to calculate a new LLRclipping value

4. Passing the new clipping value to the Sphere Detector and clip the La(c’) values

5. Repetition from 2 to 4 until the end of transmission

In [NA11] the adaptive algorithm considers the adaptation steps code block by codeblock. In our implementation, the detector proceeds frame by frame1 due to imple-mentation constraints, therefore our detection ↔ decoding iterations are made withcomplete frames. A full frame is detected and only then is deinterleaved, clipped, anddecoded.

To summarise, the purpose of this algorithm is to reduce the LLR clipping value (thusincreasing the BER), until the BER reaches the TER. It is done for a fixed SNR duringthe transmission of all the frames. If the BER of the system is already worse than theTER, the algorithm will not reduce the LLR clipping value.

4.2.1 LLR clipping value initialization

It is done only at the beginning of the algorithm. The purpose is to find an initialclipping value "so high" that the initial BER of the system (after decoding) is betterthan the TER. In scenarios where the SNR is not enough to provide a BER better thanthe TER, the adaptive clipping is useless because the L-values will never be clippedin the comparison step.

To find an adequate initialization clipping value, it must be considered that the BERafter channel decoding is expected to be even better for the frames which already meetthe TER before channel decoding. Therefore, using (2.4) to find the value of |Le(u)|,the initial clipping value can be calculated as follows:

L0cl = |Le(u)|Pb=TER = ln

(1

TER− 1

)= ln

(TER−1 − 1

)≈ − ln (TER) = LTER

(4.1)

This LLR clipping value will be applied in the External Clipping block, which is donebefore decoding. Then the BER after decoding is expected to be much better than theone intended by the initialization clipping value before decoding.

1In our simulation each frame consisted of 300 blocks. The frames are also referred as loops.

34


4.2.2 Estimation of the previous frame BER

The proposed method in [NA11] estimates the BER of the decoded blocks as anaverage of the error probability of each decoded bit using (2.4). It also proposed totake only the BER of a small amount of L-values in order to reduce the computationalcomplexity.

The selection of this subset of L-values of a decoded block is made upon their ab-solute value. It is proposed that the subset with the r -th smallest L-values is anaccurate representation of the BER of the whole block. This complexity reduction ap-proximation has been tested during this study and its validity could not been widelyconfirmed, because the dependency between the size of the subset influences too muchthe estimated BER. The following example will illustrate it:

Let the block length be 776 bits2. Then the vector Le(u) will have 776 values. LetLe(u) have 120 values with its absolute value being 0, and the rest of the elements ofthe vector (656) having non-zero values. If a subset with the smallest 503 L-values istaken, all the values of the subset will have an absolute value of 0. Therefore, accordingto [NA11], the BER of the block could be approximated as

Pb =1

776

50∑r=1

(1 + e0

)−1= 0.03222 (4.2)

Since all the L-values taken are 0, this is the BER of the worst transmission case. Tak-ing into consideration that the theoretical maximal BER4 for this calculation methodis 0.5, the method lacks of accuracy.

This inaccuracy is not a consequence of the block size used in our implementation, be-cause with the data set used in [NA11], the maximal BER would be:

Pb =1

1152

50∑r=1

(1 + e0

)−1= 0.02170 (4.3)

which is even further from 0.5.

Furthermore, this BER estimation method does not penalize the overestimated detec-tion hypotheses5, because in (2.4) a higher L-value always yields a better BER.

2This is the block length in our implementation.3Subset length used in [NA11].4The theoretical maximal BER is achieved when all the L-values are zero and the length of thesubset is equal to the length of the whole vector.

5The sphere detector is not able to find the LLR for all the bits. For those bits whose LLR valuesare not found, the SD assigns them an infinite LLR value, thus overestimating its reliability.

35


To sum up, this BER estimation method, although simple, presents three handi-caps:

• The assumption that the smallest L-values dominate the BER is not enoughaccurate because, as experienced in many simulations during this study, it iscommon that the length of the subset of L-values is not large enough to includealso non-zero L-values, therefore the BER always tends to be higher than itreally is.

• Although taking a subset of L-values to calculate the BER reduces the com-putational overload because there are less divisions, the threshold of maximalachievable BER is too jeopardized.

• It assumes that higher L-values always provide a better BER, thus becoming anerratic method when the employed LLR clipping value is greater than the onewhich fits our TER, due to the overestimation of the reliability provided by theexcessive L-values.

In our implementation the estimation of the BER is made taking all the bits, not asubset of them. In addition, our simulation setup measures the BER by frame, notby block, as it has been previously explained. Therefore, from now on all calculationreferences will be made to frames6 instead of blocks.

4.2.3 Comparison of the current frame BER with the TER to calculatea new LLR clipping value

This is the most important step of the adaptive clipping. Here the decision is made,whether the LLR clipping value is modified or not. The clipping is performed followingthe algorithm proposed in [NA11]:

L(m)cl,temp = L

(m−1)cl − µ

[ln(TER)− ln

(P

(m−1)b

)](4.4)

L(m)cl = max

{min

{LTER, L

(m)cl,temp

}, |L|min

}(4.5)

The notation used has the following meanings:

• L(m)cl,temp: Auxiliary variable to store temporarily the candidate clipping value for

the frame m7.6We refer to information blocks as frames.7In [NA11] m refers to block instead of frame.

36


• L(m)cl : Clipping value of the frame m.

• P (m−1)b : BER of the previous frame.

• LTER: The initialization clipping value calculated in (4.1).

• |L|min: Minimum possible LLR magnitude determined by the fixed point accu-racy8.

• µ: A step value. The higher this value is, the faster the adaption is made9.

In (4.4) the logarithm of the previous frame BER is subtracted from the TER andmultiplied by a scaling value µ. The result is subtracted from the clipping value ofthe previous frame. If the result of the subtraction from the logarithmic operations ispositive, that means the BER of the previous frame is better (lower) than the TER,therefore the clipping value is reduced.

In (4.5) the unnecessary L(m)cl increase is avoided using the min operation. When

LTER is chosen over L(m)cl it means that the best achievable BER is worse than the

TER, therefore it is not clipped. With the max operation L(m)cl values smaller than the

available accuracy of the fixed point representation are avoided, which is not relevantin our implementation because floating point has been used.

4.2.4 Passing the clipping value to the Sphere Detector and clip theLa(c’) values

Once the Lcl value for the next frame is obtained, it is passed to the External Clippingblock and to the Sphere Detector.

The External Clipping block clips each value of the vector La(c’) with the receivedLcl. Therefore, after the external clipping:

− Lcl ≤ La,cl(c’) ≤ +Lcl (4.6)

In the Sphere Detector the new Lcl value will be used as the new radius clippingvalue. As it has been explained in Chapter 2, this operation is referred as internalclipping.

Using the same internal and external clipping value is useful because the radius clip-ping consists of the elimination of counter-hypotheses paths10 which will be further

8In our implementation floating point is used, therefore this value is 0.9In our implementation a value of 0.1 is used, following the indications of [NA11].

10Referred to the tree-search-based sphere detector introduced in Chapter 2.

37

4.3. RESULTS

clipped [Zim07] [Ade09]. For those paths whose metrics are larger than the radiusclipping level, no counter-hypotheses are found because they are not explored. It doesnot matter which L-values would have been obtained with this counter-hypothesesbecause they would not have been taken into account after the external clipping.Therefore, using an internal clipping larger than the external one only adds redun-dancy.

4.3 Results

To analyse the proposed adaptive algorithm, the depicted simulation in Figure 4.4 hasbeen performed with the following setup:

• MIMO 4x4 64 QAM PCCC Decoder

• TER: 10−2

• SNR: 20 dB

In a first overview, it is noticeable the offset of approximately three orders of magnitudebetween the estimated BER and the real BER, because in our implementation theBER is estimated from Le(c’) instead of Le(u). Therefore TER employed in thealgorithm has to be worse than the desired real TER. It is proposed as future workto analyse the abovementioned offset in order to establish its behavior for differentsystem configurations.

38

4.3.RESU

LTS

0 50 100 150 200 250 300 350 400 450 50010 10

10 9

10 8

10 7

10 6

10 5

10 4

10 3

10 2

10 1

100

BER

0 50 100 150 200 250 300 350 400 450 5000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

LLR

Clip

ping

Val

ue

Frame Transmitted

LLR Clipping ValueEstimated BEREstimated BER (moving average)Real BERReal BER (moving average)

Initialization Clipping Value:Lcl,0= ln(TER)=4.6

Adapted LLR Clipping Value 3.3 BER Offset

Estimated BER

Real BER

Figure 4.4: Adaptive LLR Clipping Simulation for 4x4 64 QAM PCCC 1 int.it. 1 ext.it., SNR=20 dB, TER=10−2

39

4.3. RESULTS

Figure 4.4 contains a representation of the LLR clipping value adaptation vs. theBER performance. The graphic has two different vertical axis: On the left there isa logarithmic BER scale and on the right there is a linear LLR clipping value scale.The abscise axis counts the loops performed. In each loop a frame is transmitted.The dotted blue line represents the estimated BER of each frame, which is averaged(solid blue line) using the moving average method, taking into account the previous50 values. Note that the estimated BER is the one used in the algorithm equations(4.4). Additionally, the brown lines display the real11 BER. The green line representsthe LLR clipping value for each transmitted frame.

The initialization clipping value Lcl,0 = 4.6 is pointed in the graphic. If the readercompares the real BER in this figure with the correspondent real BER depicted inAppendix A.3 for the same configuration for 20 dB, it is observable that the real BERin Figure 4.4 is almost the same because the initial and the adapted clipping value arevery similar to the one used in the case study (4.4 ).

As it is expected, when the clipping value is reduced, the BER of the frames areincreased, and therefore, with some delay, the moving average of the BER varies.Note that the reduction of the clipping value is produced approximately in a linearfashion. The factor that determines the slope of the adaptive clipping line (whichdenotes the clipping speed) is the difference in each loop between the TER and thecurrent BER.

When the estimated BER reaches the TER, the LLR clipping value oscillates arounda stable value (the Adapted LLR Clipping Value) and the average BER remains atthe TER. An analysis of how the BER offset differs for different system configurationsis essential in order to set an estimation TER12 that yields a real BER close to thereal desired TER. It is therefore proposed as future work.

An adaption time can be deduced from the Figure 4.4 by means of the number of framesneeded to achieve a stable LLR clipping value or, equivalently, the TER. Having 500transmitted frames in total, we needed 37 transmitted frames (7.4%) to adapt theclipping value. In a real transmission the amount of frames transmitted tends to bevery high or infinite, therefore the amount of time needed to adapt the clipping valuewould become insignificant. Although different step values (µ) have not been tested,the step µ = 0.1 proposed in [NA11] has shown a good performance because theconvergence to the TER has been fast.

11Not an estimation, but the ratio between the error bits after decoding and the number of transmittedbits. Note also that the real BER (dotted brown line) lacks of many values— In that cases theBER is 0.

12The one used in the algorithm and the one to which we refer when we write TER alone.

40

4.3. RESULTS

Finally, when choosing the TER is important to consider the frame length becauseit defines a TER threshold. E.g.: If a frame has 116400 bits (our case), the worstmeasurable BER is 1

116400 = 8.59 ·10−6. Thus, setting a TER smaller than that wouldhave no effect. The same applies to the block size if the adaption is made block byblock.

It can be concluded that the adaptive approach reduces the performance predictionburden needed by the mutual information exchange approach because the LLR clippingvalue needed to achieve the TER can be found after just a few transmitted frames.However, this algorithm has the deficiency that it does not take into account theoverestimation of the detection hypotheses, because considers that a higher L-valueyields always a better BER, which is not correct as it was already introduced. Ananalysis of how this disadvantage can affect the behavior of the adaptive clipping isproposed as future work.

41

5 Summary

By means of this study several communication system configurations (varying thenumber of transmit and receive antennas, constellation size, decoder type, itera-tions...) have been examined through Matlab simulations in order to find performance-complexity trade-offs. The great number of simulations has output rich-data chartsand graphic representations from where the results can be interpreted and the featuresof the configurations can be easily recognized.

Additionally, an implementation of an adaptive LLR clipping algorithm has been pro-posed to adaptively track the BER and adapt it to the TER by means of the LLRvalues, taking advantage of the detection ↔ decoding scheme.

5.1 Outcome

The chief contribution of this work is the outcome provided by the variety of com-munications system configurations simulated. The data charts and graphics rep-resentations have yielded valuable results, from which the following can be high-lighted:

• An approximate relationship between the number of internal and external iter-ations needed to achieve a BER of 10−5 has been found, valid for all the consid-ered system configurations. For any number of external iterations, 3-6 internaliterations in PCCC and 7-8 in LDPC provide the best SNR performance.

• It has been quantified that the maximum achievable SNR gain using LDPCreaches a 90% with 2 external iterations while PCCC needs 2 external iterationsto achieve a 70% and 3 to achieve a 90%. However, the overall performanceof LDPC is worse than PCCC because requires a higher number of internaliterations for the same BER performance.

• The benefit of iterative detection↔ decoding is not very significative in systemswith big constellation sizes (64 QAM) but small number of antennas (2x2),being this particularity even worse when using LDPC instead of PCCC. Unless

43

5.2. FURTHER WORK

the latency were a critical parameter, to use MIMO 4x4 would provide betterspectral efficiency as well as BER performance.

• A fixed proportion between the number of extended nodes in 16 QAM and64 QAM has been found. The 4x4 16 QAM system has always an approximately90% of the number of nodes extended by the 4x4 64 QAM system for 1 externaliteration, and 80% for 2 to 4 external iterations. The percentage does not varywith the number of internal iterations and can be applied to both the PCCCand the LDPC case.

• The SNR gain in iterative systems is smaller for smaller systems. The SNRimprovement between 16 QAM and 64 QAM in iterative cases is between 0.1 dBand 0.7 dB (higher values for higher number of external iterations) using PCCCand an almost constantly 0.6 dB using LDPC.

• Using LDPC less nodes are extended in all the cases, except for small systems(2x2), where it has a similar complexity as PCCC. For systems with an increas-ing number of both internal and external iterations, this complexity differencebecomes higher.

As a certain BER performance can be achieved with more than one configuration(different number of antennas, constellation size, decoder type, internal and externaliterations), the best trade-off between performance and complexity depends eventuallyon the application scenario.

Furthermore, the case study constitutes a valuable resource for reference. E.g.: Itallows to know, by using the appropriate graphic, what is the BER for a wide rangeof SNR values in a given configuration.

Finally, an implementation of the adaptive LLR clipping algorithm from [NA11] hasbeen made. With it, the BER performance is tracked and adapted to the TER bymeans of the LLR values. Using this approach, the previous mutual information simu-lation is not needed to provide an LLR clipping value that fits our TER and thereforeit allows to calculate the clipping value "on the fly", reducing the computational timeof the simulation system.

5.2 Further work

The analysed case study is open to all kind of additional contributions because itcontains a broad range of parameters which can be further studied and modified tolook for performance-complexity trade-offs. Therefore:

44

5.2. FURTHER WORK

• A continuation on the work of obtaining results for different number of antennas,constellation sizes and decoder types can be valuable.

• It is proposed to extend the investigation of trade-offs in performance-complexityfor scenarios with parameter variations not considered in this study. E.g.: Fordifferent channel models, coding rates, block sizes, equalization strategies, etc.

• The inclusion of the adaptive clipping algorithm in the case study has not beenconsidered. To simulate the different configurations with the adaptive clippingapproach may improve the trade-off cases.

• To analyse the behavior of the BER offset between the set TER and achievedTER in the LLR adaptive clipping is important.

• In general, the iterations, both internal and external, have played an impor-tant role in this study. To investigate further adaptive schemes where theperformance-complexity trade-offs are enhanced iteration by iteration would bean interesting research topic.

45

A Performance-Complexity Data Charts

In the following pages, the data charts containing the results of the different simulatedconfiguration systems will be shown. For the correspondent graphic representationsplease see Appendix B.

47

A.1. MIMO 2X2, 64 QAM, PCCC DECODER

A.1 MIMO 2x2, 64 QAM, PCCC Decoder

int.it. ext.it.Eb/N0 at

BER≈10e-5Complexity

SNRImprovement

ComplexityImprovement

1 1 ~ 19,2 dB 91 2 ~ 17,4 dB 20 ~ 1,8 dB x 2,21 3 ~ 17,1 dB 31 ~ 2,1 dB x 3,41 4 ~ 17,1 dB 42 ~ 2,1 dB x 4,72 1 ~ 17,3 dB 92 2 ~ 15,7 dB 20 ~ 1,6 dB x 2,22 3 ~ 15,7 dB 31 ~ 1,6 dB x 3,42 4 ~ 15,6 dB 42 ~ 1,7 dB x 4,73 1 ~ 17,0 dB 93 2 ~ 15,4 dB 20 ~ 1,6 dB x 2,23 3 ~ 15,5 dB 31 ~ 1,5 dB x 3,43 4 ~ 15,3 dB 42 ~ 1,7 dB x 4,74 1 ~ 16,9 dB 94 2 ~ 15,3 dB 20 ~ 1,6 dB x 2,24 3 ~ 15,3 dB 31 ~ 1,6 dB x 3,44 4 ~ 15,3 dB 43 ~ 1,6 dB x 4,85 1 ~ 16,8 dB 95 2 ~ 15,3 dB 20 ~ 1,5 dB x 2,25 3 ~ 15,3 dB 31 ~ 1,5 dB x 3,45 4 ~ 15,2 dB 42 ~ 1,6 dB x 4,76 1 ~ 16,7 dB 96 2 ~ 15,2 dB 20 ~ 1,5 dB x 2,26 3 ~ 15,3 dB 31 ~ 1,4 dB x 3,46 4 ~ 15,2 dB 42 ~ 1,5 dB x 4,77 1 ~ 16,7 dB 97 2 ~ 15,1 dB 20 ~ 1,6 dB x 2,27 3 ~ 15,3 dB 31 ~ 1,4 dB x 3,47 4 ~ 15,2 dB 42 ~ 1,5 dB x 4,78 1 ~ 16,5 dB 98 2 ~ 15,1 dB 20 ~ 1,4 dB x 2,28 3 ~ 15,2 dB 31 ~ 1,3 dB x 3,48 4 ~ 15,3 dB 42 ~ 1,2 dB x 4,79 1 ~ 16,6 dB 99 2 ~ 15,2 dB 20 ~ 1,4 dB x 2,29 3 ~ 15,1 dB 31 ~ 1,5 dB x 3,49 4 ~ 15,1 dB 42 ~ 1,5 dB x 4,710 1 ~ 16,5 dB 910 2 ~ 15,1 dB 20 ~ 1,4 dB x 2,210 3 ~ 15,2 dB 31 ~ 1,3 dB x 3,410 4 ~ 15,1 dB 42 ~ 1,4 dB x 4,7

48

A.2. MIMO 2X2, 64 QAM, LDPC DECODER

A.2 MIMO 2x2, 64 QAM, LDPC Decoder



SNRImprovement



49





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50





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51





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52





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53

B Performance-Complexity Graphics

In this appendix are shown the graphics that combine the 5 performed external itera-tions for each number of internal iterations, for each system configuration. The datasource for this graphics is displayed in Appendix A. The graphics on the left repre-sent the BER performance, while the graphics on the right represent the complexity(measured as the number of extended nodes).

Each graphic contains the representation of the results from 1 to 5 external iter-ations, being the outcome with 1 external iteration the curve in the lightest bluecolor, and increasing the darkness of the curves with an increasing number of exter-nal iterations. Each row of graphics show the result for a specific number of internaliterations.

B.1 MIMO 2x2, 64 QAM, PCCC Decoder

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Student Research Project-Raul Aviles Poblador

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